As to the secondary question, the character was named Robin because he was modeled on Robin Hood.
Dick Grayson and the Robin persona were created by Bob Kane, Bill Finger, and Jerry Robinson. He was introduced as “Robin the Boy Wonder” and took visual cues from the acrobats of the day and Robin Hood the legendary character.
Jerry Robinson himself said:
I came up with Robin Hood because The Adventures of Robin Hood were boyhood favorites of mine. I had been given a Robin Hood book illustrated by N. C. Wyeth ... and that's what I quickly sketched out when I suggested the name Robin Hood, which they seemed to like, and then showed them the costume. And if you look at it, it's Wyeth's costume, from my memory, because I didn't have the book to look at.
- The Comics Journal #271
N.C. Wyeth's Robin Hood, left; Robin's first appearance, right
At the time Robin first appeared in 1940, the "Silver Age" convention of making sidekicks a miniature version of the superhero - e.g., "Aqualad", "Supergirl", etc - hadn't happened yet, so there was no real reason to name the character "Batboy".